On the other hand, the theory of random walks, also known as Brownian motion in the scaling limit, plays an essential role in many stochastic processes and challenging problems in probability and statistical physics [1]}, [2]}, [3]}. In two dimensions, random walks and percolation at its critical point share various interesting properties other than their conformal invariant property. For example, the external perimeters of a critical percolation cluster and a planar random walk in the continuum limit are believed to be described by the scaling limit of planar self-avoiding random walks (SAWs) [4]}. Moreover, the trace of a smart kinetic self-avoiding random walk [5]} (i.e., the trace of a random walker that never crosses itself and never gets trapped) produces statistically the same fractal path in accord with the perimeter (or the hull) of a critical percolation cluster in 2D.
